Horizontal Curves - Theory & Concepts

Horizontal Curves - Theory & Concepts

Detailed study of simple, compound, reversed, and spiral curves used in route surveying.

Overview of Horizontal Curves

Horizontal curves provide a smooth, gradual transition between two intersecting straight lines (tangents) in the horizontal plane of a route alignment. They are primarily designed as circular arcs of specific radii, ensuring safe and comfortable vehicle operation at designed speeds.

Types of Horizontal Curves

Curve Classifications

Simple Curve: A single circular arc connecting two tangents.
Compound Curve: Two or more circular arcs of different radii turning in the same direction, with a common tangent and point of compound curvature (PCC).
Reversed Curve: Two circular arcs turning in opposite directions, with a common tangent at the point of reversed curvature (PRC).
Spiral (Transition) Curve: A curve with a continuously changing radius, providing a gradual transition from a straight tangent (infinite radius) to a circular curve of constant radius.

Elements of a Simple Horizontal Curve

Key Points and Terminology

PC (Point of Curvature): The beginning of the curve, where the tangent ends.
PT (Point of Tangency): The end of the curve, where the tangent begins.
PI (Point of Intersection): The point where the back tangent and forward tangent intersect.
I (Intersection Angle): The central angle subtended by the curve, equal to the angle of deflection at the PI.
R (Radius): The radius of the circular curve.
T (Tangent Distance): The distance from PC to PI, or PI to PT.
L (Length of Curve): The length of the circular arc from PC to PT.
E (External Distance): The distance from the PI to the midpoint of the curve.
M (Middle Ordinate): The distance from the midpoint of the long chord to the midpoint of the curve.
C (Long Chord): The straight-line distance from PC to PT.

D

Defining the Sharpness of a Curve

The Degree of Curve (

D

) defines the sharpness of the curve. There are two standard definitions:

Arc Basis (Metric Standard): The central angle subtended by a $20 \text{ m}$ (or $100 \text{ ft}$ ) arc.
Chord Basis (Railway Standard): The central angle subtended by a $20 \text{ m}$ (or $100 \text{ ft}$ ) chord.

$$
D = \\frac{1145.916}{R} \\text{ (Metric, 20 m arc)}
$$

Important

For the US Customary system using a

100 \text{ ft}

arc, the constant is

5729.58

. The relationship is strictly derived from the arc length formula:

\frac{D}{20} = \frac{360}{2\pi R}

for metric or

\frac{D}{100} = \frac{360}{2\pi R}

for English.

Fundamental Geometric Formulas

$$
T = R \\tan\\left(\\frac{I}{2}\\right)
$$

$$
L = \\frac{I}{D} \\times 20 \\text{ (Metric)} \\quad \\text{or} \\quad L = \\frac{\\pi R I}{180}
$$

$$
C = 2R \\sin\\left(\\frac{I}{2}\\right)
$$

$$
E = R \\sec\\left(\\frac{I}{2}\\right) - R = R \\left[ \\sec\\left(\\frac{I}{2}\\right) - 1 \\right]
$$

$$
M = R - R \\cos\\left(\\frac{I}{2}\\right) = R \\left[ 1 - \\cos\\left(\\frac{I}{2}\\right) \\right]
$$

Stationing Computations

Calculating Route Stations

Route alignment is tracked using stationing. The station of the PC is found by subtracting the Tangent distance from the PI station. The station of the PT is found by adding the Curve Length to the PC station. The PI station is never calculated by going backwards from the PT.

$$
\\text{Sta. PC} = \\text{Sta. PI} - T \\quad \\text{and} \\quad \\text{Sta. PT} = \\text{Sta. PC} + L
$$

Field Layout: Deflection Angles and Chords

Setting Out the Curve

To lay out a simple curve in the field using a total station or theodolite set at the PC, the curve is divided into smaller segments. A sub-chord (

c

) is used to connect full stations. The deflection angle (

\delta

) from the tangent to any point on the curve is half the central angle (

\theta

) subtended by the arc to that point.

Because full stations rarely fall exactly on the PC and PT, surveyors must calculate an initial sub-chord from the PC to the first full station, full chords between intermediate stations, and a final sub-chord from the last station to the PT.

Deflection Angle for any Sub-chord ( $c$ )

$$
\\delta = \\frac{c \\times D}{2 \\times 20} \\text{ (Degrees, using 20m arc)}
$$

Where:

$\delta$ = Deflection angle for the sub-chord (degrees)
$D$ = Degree of curve (based on 20 m arc)
$c$ = Length of the sub-chord or full chord (m)

Note

The total deflection angle to the PT must equal exactly

I / 2

. This serves as a critical field check to ensure all intermediate deflection angles were computed correctly.

Sight Distance on Horizontal Curves

Middle Ordinate Clearance

To ensure safe stopping sight distance (

S

) on a horizontal curve, a clear line of sight must be maintained across the inside of the curve. If the sight distance is less than the curve length (

S < L

), the minimum clearance distance from the centerline of the inside lane to an obstruction (like a wall, building, or cut slope) is defined by the middle ordinate (

M

$$
M = R \\left[ 1 - \\cos\\left(\\frac{28.65 S}{R}\\right) \\right]
$$

Where:

$M =$ Middle ordinate clearance distance from the centerline of the inside lane (m)
$R =$ Radius of the curve to the centerline of the inside lane (m)
$S =$ Required sight distance (m)

Setting Out Simple Curves

Deflection Angle Method

The most common field method for setting out (staking) a simple curve is the deflection angle method using a total station or theodolite set up at the PC.

The deflection angle to any point on the curve is equal to half the central angle subtended by the arc from the PC to that point. The surveyor turns the calculated cumulative deflection angle from the tangent line and measures the corresponding chord distance to set the stake.

Offset Methods

When instruments are unavailable or precision requirements are low, curves can be set out using linear measurements alone:

Offsets from the Tangent: Perpendicular distances ( $y = R - \sqrt{R^2 - x^2}$ ) are measured from established points ( $x$ ) along the tangent line to locate points on the curve.
Offsets from the Long Chord: Perpendicular offsets are measured from the long chord connecting the PC and PT to locate curve points.

Compound and Reversed Curves

Compound Curves

Compound curves involve two circular curves (

R_1

and

R_2

) meeting at a Point of Compound Curvature (PCC). The centers of the two curves lie on the same side of the common tangent passing through the PCC.

$$
I = I_1 + I_2 \\quad \\text{and} \\quad t_1 = R_1 \\tan\\left(\\frac{I_1}{2}\\right), \\, t_2 = R_2 \\tan\\left(\\frac{I_2}{2}\\right)
$$

Where:

$I =$ Total intersection angle
$I_1, I_2 =$ Central angles of the first and second curves
$t_1, t_2 =$ Tangents of the individual curves from the common tangent

Note

The length of the common tangent is the sum of the individual tangents:

T_c = t_1 + t_2

. The total tangent distances from the main PI to the PC (

T_1

) and from the main PI to the PT (

T_2

) can be found by solving the triangle formed by the PI and the two intersection points on the common tangent using the sine law.

Reversed Curves

Reversed curves involve two circular arcs turning in opposite directions meeting at a Point of Reversed Curvature (PRC). The centers of the curves lie on opposite sides of the common tangent. They are generally avoided on high-speed highways due to the sudden change in centrifugal force and superelevation requirements, but are common in railway switchbacks and slow-speed urban roads.

Parallel vs. Non-Parallel Tangents

Parallel Tangents: If the initial back tangent and final forward tangent are parallel, the central angles of both curves are exactly equal ( $I_1 = I_2$ ). The perpendicular distance ( $d$ ) between the tangents is $d = R_1(1 - \cos I_1) + R_2(1 - \cos I_2)$ .
Converging/Diverging Tangents: The total angle between the main tangents is determined by the difference or sum of the individual intersection angles depending on geometry.

Spiral (Transition) Curves

Purpose of Transition Curves

Spiral curves are introduced between straight tangents and circular curves to provide a gradual introduction of centrifugal force and superelevation. The most common type is the clothoid (Euler spiral), where the radius decreases linearly as the length along the curve increases (

R \times L = \text{constant}

Elements of a Spiral Curve

Key Elements

TS (Tangent to Spiral): The point where the straight tangent ends and the spiral begins.
SC (Spiral to Circular): The point where the spiral ends and the circular curve begins.
CS (Circular to Spiral): The point where the circular curve ends and the second spiral begins.
ST (Spiral to Tangent): The point where the second spiral ends and the straight tangent begins.
$L_s$ : Length of the spiral from TS to SC (or CS to ST).
$\theta_s$ : Spiral angle, the total angle turned through by the spiral.

$$
\\theta_s = \\frac{L_s}{2 R_c} \\text{ (radians)} \\quad \\text{or} \\quad \\theta_s = \\frac{180 L_s}{2 \\pi R_c} \\text{ (degrees)}
$$

Where:

$L_s =$ Length of the spiral
$R_c =$ Radius of the central circular curve

$$
T_s = (R_c + p) \\tan\\left(\\frac{I}{2}\\right) + k
$$

Where:

$T_s =$ Total tangent distance from PI to TS (or PI to ST)
$R_c =$ Radius of the circular curve
$I =$ Total intersection angle
$p =$ Shift of the circular curve inward
$k =$ Distance from TS to the point opposite the shifted center

Note

For many practical purposes, the values of

p

and

k

can be approximated as:

p \approx \frac{L_s^2}{24 R_c}

and

k \approx \frac{L_s}{2}

Key Takeaways

Horizontal curves (Simple, Compound, Reversed, Spiral) safely connect intersecting tangents.
A Simple Curve is defined by its Radius ( $R$ ), Intersection Angle ( $I$ ), and Degree of Curve ( $D$ ).
The primary metric formulas involve Tangent ( $T = R \tan(I/2)$ ), Length ( $L = \pi R I / 180$ ), and Chord ( $C = 2R \sin(I/2)$ ).
Setting out simple curves is most commonly done using the Deflection Angle Method from the PC, ensuring the final angle equals $I/2$ .
Spirals are critical on high-speed routes to gradually introduce curvature and banking (superelevation), preventing sudden lateral forces.

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Horizontal Curves - Theory & Concepts

Overview of Horizontal Curves

Types of Horizontal Curves

Curve Classifications

Elements of a Simple Horizontal Curve

Key Points and Terminology

Degree of Curve (DDD)

Defining the Sharpness of a Curve

Important

Fundamental Geometric Formulas

Stationing Computations

Calculating Route Stations

Field Layout: Deflection Angles and Chords

Setting Out the Curve

Note

Sight Distance on Horizontal Curves

Middle Ordinate Clearance

Setting Out Simple Curves

Deflection Angle Method

Offset Methods

Compound and Reversed Curves

Compound Curves

Note

Reversed Curves

Parallel vs. Non-Parallel Tangents

Spiral (Transition) Curves

Purpose of Transition Curves

Elements of a Spiral Curve

Key Elements

Note

Degree of Curve ( $D$ )